Using the Fisher information matrix, one can obtain the instrumental uncertainty of the measurement of GW LD. It is assumed that the LD (\(D_{\rm L}\) ) is uncorrelated with the errors on the remaining GW parameters (the inclination angle \(\iota =0\) ), \(\mathcal {H}\propto D_{\rm L}^{-1}\) , and the corresponding instrumental uncertainty can be written as, \(\sigma _{D_{\rm L}}^{\rm inst}\simeq {2D_{\rm L}/ {\rho }}\,\) . Furthermore, the LD is also affected by the effect of the weak lensing, and the lensing uncertainty can be assumed as \(\sigma _{D_{\rm L}}^{\rm lens}=0.05z D_{\rm L}\)  [1]}, [2]}, [3]}. Thus, the total uncertainty of GW LD measurements is taken to be,

DL=(DLinst)2+(DLlens)2
